In the above thread there are seven in total different occurrences of these highly unusual contrails captured on photo or video. I have trawled Google Images, flickr and youtube to find more reported occurrences and raised the total to more than 60 cases spanning over the past 15 years. One example has already made to the Wikipedia article on Contrail (English and Polish versions).

Sinuous vapor trail over south-east Poland.

Here is a selection of observed thin contrails (in three galleries):

These numerous examples show that sinusoidal thin contrails are not a freak event but a recurrent albeit rare phenomenon. They are exactly what they look like - solitary hybrid contrails. Hybrid contrails are (parts of) exhaust contrails entrained in the wing tip vortices: http://contrailscience.com/hybrid-contrails-a-new-classification/. Normally, hybrid contrails come in pairs, like the vortices themselves. They are formed in a narrow range of RH and usually observed behind large planes that create stronger wake vortices, e.g., Boeing 747. The lifetime of a hybrid contrail pair is limited by the Crow instability (http://en.wikipedia.org/wiki/Crow_instability) that breaks down a pair of counter rotating vortices. Formation of solitary hybrid contrails apparently occurs in the same range of RH but additionally requires some form of atmospheric turbulence capable of decoupling the vortex pair. Solitary vortices can have a longer lifetime, therefore solitary hybrid contrails can persist, unlike a hybrid contrail pair.

There are instabilities in solitary vortices causing vortex meandering, a phenomenon which has been observed in experimental studies of airfoil tip vortices in wind tunnels.

Solitary hybrid contrails display similar behaviour: in the beginning they form fairly straight lines and then transform into meandering shapes of gradually increasing amplitude. This phenomenon yet awaits its researcher, but the underlying causes are likely to be the same or similar to the causes of the Crow instability: an ambient turbulence and self induction of a vortex with itself. Playing on the name of its discoverer, I propose the name for the solitary vortex instability: the 'Rook' instability. Rook is a bird of the corvid family; the juvenile rook is similar to carrion crow, whereas their adults are readily distinguishable.

Alternative explanation of the sinusoidal shape, as suggested in several discussions of individual occurrences of solitary thin contrails, is their perturbation by atmospheric waves. This explanation however can be ruled out by the observations of a sinusoidal thin contrail coexisting with an ordinary exhaust contrail from the same plane which is not perturbed (see the last gallery above).

Below is a published account of one of the observation (from onlinelibrary.wiley.com/doi/10.1256/wea.266.05/pdf)

Contrail contortions

I wonder if any of your readers are able to offer an explanation for the unusual behaviour of an aircraft contrail that I observed a few days ago?

On 19 November 2005, at exactly noon, I noticed the contrail of a large twin-engined jet, most likely a Boeing 777 or an Airbus A330, heading eastwards over my home in Reading (and then Heathrow), at high altitude. Judging by other aircraft contrails that it crossed, I would guess it to be at about 27 000 to 30 000 ft (8000 to 9000 m) as other craft were certainly higher. The weather at ground level was dead calm, frosty and sunny.

What drew my attention initially to this aircraft was that it produced two distinctly separate contrails, that from the port engine having the usual puffy diffuse appearance, while the starboard trail apparently formed a very tight vortex, which initially drew a razor sharp line behind the starboard engine, only slightly offset towards the wingtip. The port contrail dispersed and disappeared within about 30 seconds (say 5 km behind the plane), while the starboard contrail was highly persistent, lasting over 10 minutes before finally disappearing without spreading at all. However, the most noticeable feature after the first minute was the slow formation of highly pronounced sinuous contortions, as clearly shown in Fig 1(a). It was hard to judge the alignment of the contortions, or even whether they were in a single plane, though they were certainly not helical, and not obviously rotating as in a vortex. Mostly the contrail itself persisted as a very thin, sharply defined line, but at just a few places there were anomalies looking like stray tufts of wool from a homespun thread Fig 1(b). These did not rotate even though they looked funnel shaped.

Unfortunately, I was too slow fetching my camera to be able to capture the early stages of the phenomenon, while there were still two contrails, but I was easily able to photograph the latter stages (Fuji Finepix 2800 digital camera mainly on 38mm zoom, but with some at 4 to 6 times zoom). Although I watched another 20 or more aircraft contrails during the following hour, none of them showed any similarly unusual behaviour.

I would be most interested to hear if anyone else has made similar observations, or can explain the phenomenon.

Solitary hybrid contrails compared to ordinary hybrid and exhaust contrails of similar ages:

Solitary hybrid contrails in 3D

In at least two cases, the solitary hybrid contrails have been observed from different locations and photographed at about the same time. Using these photographs I have reconstructed the contrails' shapes in the form of pseudo stereo images:

Rotterdam, February 18, 2013

Göttingen, January 22, 2014

Neither of these two contrails lays in one plane. Both consist of a number of irregular helical turns, the shapes of which resemble a stretched paperclip rather than a coil spring. Viewed at some angles, such irregular helical shapes may look very different from a sine curve, like, for example:

and

Appeal to the readers:

A solitary hybrid contrail is a very rare phenomenon. Therefore, if you happen to see one, please observe it thoroughly. Please note the time and place of observation and report it here asap. This would help the identification of atmospheric conditions as well as the types of planes and would facilitate further research of this phenomenon.

The leading position of UK is likely due to a bias of my searches toward English compared to other languages that I've searched with. However, the European dominance over the rest of the world is significant. The reasons for this is yet unknown. It could be due to an increased occurrence of clear air turbulence that correlates with the position of jet stream and/or a higher density of large aircraft routes over central Europe.

I think it's a combination of factors....but you're on the right track.....

The "sinusoidal" element that causes "some" persistent contrails to develop in that visible way is likely due to atmospheric processes....partly as a result of the way the airplane can affect the air "currents" as it passes (**)....and also how the existing atmospheric forces, PRIOR to when the airplane entered, are going to interact.

The airplane can be a "trigger" in this interpretation. Of course, "Mother Nature" uses other "triggers", long before airplanes were invented by Humans.

Also....since clear air is (virtually) transparent to Human eyes. When clouds form it tends to "draw attention". But the currents....whether horizontal, vertical or a combination? Of "clear" air are invisible...we only see the effects of such currents ON the clouds, which are "visible"...this includes contrails, of course.

A helix is sinusoidal, just in three dimensions. The point is more that it's a 3D squiggle, not a 2D squiggle.

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A helix is not sinusoidal. The projection of a helix may be sinusoidal. The word sinusoidal imples a 2D curve lying in a plane.
The point of my question was whether this contrail lies in a plane or is a 3D helix.

A helix is not sinusoidal. The projection of a helix may be sinusoidal. The word sinusoidal imples a 2D curve lying in a plane.
The point of my question was whether this contrail lies in a plane or is a 3D helix.

However looking at all the examples above, they seem (as Trailspotter noted) rather flat. Just not always entirely flat - and this variation can give some trails that look odd from some angles, especially this example:

With that one you have to think how it could be laid out on a flat surface with none of the "loops" actually crossing. Then perspective and some vertical displacement gives the illusion of loops.

In reality, or more accurately, in my 3D reconstructions, the helical turns are rather flat. A good approximation of the contrail shape is a wire loosely wrapped around school ruler:
viewed from a flat side it is sinusoidal.

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What makes it flat?
I also wonder what determines the direction of the flat side. Is it usually vertical, horizontal, or random direction?

BTW does anyone know a meaningful explanation of the Crow instability? The Wikipedia description is obscure to me.

BTW does anyone know a meaningful explanation of the Crow instability? The Wikipedia description is obscure to me.

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The simplest explanation is that two parallel vortices that rotate in opposite directions in a fluid are inherently unstable, and will interact with each other - magnifying existing irregularities in each other.

All planes have these two wake vortices, regardless of if you can see them. So they all have these squiggles in the air behind them. You just only see them when a hybrid contrail forms in them.

What makes it flat?
I also wonder what determines the direction of the flat side. Is it usually vertical, horizontal, or random direction?

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This is a good question, to which I know no answer. Finding it would require more observational data and a professional research. You can get some idea about these from the Crow's original article (a free copy of which is available at http://flair.monash.edu.au/seminars/2007/Crow_1970.pdf).

Above The Hague (Netherlands) (very near the royal stables) looking due west on 18-02-2015 around 16:00
There was a contrail like the ones in this forum. Very sharp, and curvy. It reminded me of a rocked fail.
I did not take a picture and the trail disappeared quickly in about 10 minutes after I had first seen it the trail was almost gone. Looking at google maps I figure this contrail must have been above sea near Hoek van Holland
Hope this helps.

Above The Hague (Netherlands) (very near the royal stables) looking due west on 18-02-2015 around 16:00
There was a contrail like the ones in this forum. Very sharp, and curvy. It reminded me of a rocked fail.
I did not take a picture and the trail disappeared quickly in about 10 minutes after I had first seen it the trail was almost gone. Looking at google maps I figure this contrail must have been above sea near Hoek van Holland
Hope this helps.

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Also spotted by Willem, who emailed me this, with pics

Location: Delft, The Netherlands (https://goo.gl/maps/QyWEX position is accurate to within about 2 meters)
Time: 16:03 CET
Direction of the trail: About east-west (or vice versa; I didn't see a plane on either side of the trail), but since it's hard to guess directions, it may have been more towards east southeast-west northwest.
Duration: I didn't think to time it, but it was gone relatively quickly. I think within 5 to 10 minutes.
Photo's: taken with my iPhone while trying to avoid exposing the camera to the sun. So the quality is iffy and the angles are weird :).

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CET is UTC+1, so 16:00 CET is 15:00 UTC
Location is 52.0094494,4.3354893

Thanks to @FrankN and Wilem for this observation. I will look into it this weekend, after I return home from traveling. I've checked FR24, there were a few planes above the area at the time. My favourite candidate is SIA308 Airbus A380 that flew over it ten minutes earlier, at about 14:50 UTC.

My favourite candidate is SIA308 Airbus A380 that flew over it ten minutes earlier, at about 14:50 UTC.

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That certainly matches Willem's viewpoint in the park (note I copied the wrong coordinates, which should have been 52.0094494,4.3354893, just a few hundred feet away though), and is looking towards the Hook of Holland, as @FrankN reported.

Another photo has emerged from the same day, but nearly two hours later.

Photo: Tim BAETENS, (c) CAELESTIA/Belgisch UFO-meldpunt

In this case, the location and direction can be more accurately determined, as around 51° 2'28.14"N, 3°43'36.20"E, about at the intersection of Kantienberg and Kramersplein, taken from inside a car.

A likely candidate is RBA98, a 787 from London to Dubai (kml attached)

In this case it was climbing out of London, not descending towards it, but would still have been at around 34,000 feet over the coast, and then 35000 when overhead. The apparent angle above the building puts it between the coast and Bruges. So would have been quite visible in this region:

The drift of the trail from the radar track matches the wind direction from the earth.nullschool.net images above.

Attached Files:

I found a series of photos by A380spotter depicting an interesting case of contrail evolution. Two Cargolux B747s, LX-VCA and LX-SCV, crossed their paths over Heathrow on April 6, 2015 at about 16:00 BST (15:00 UTC):

Each plane left a pair of hybrid contrails:

However, the evolutions of these pairs were different. In one pair the trails dissipated simultaneously through the Crow instability, whereas in the other pair one of the trails dissipated quicker than the other:

Unfortunately, there are no further pictures featuring evolution of the remaining solitary thin trail, that I believe would have got squiggly. Also, unfortunately, I found this series too late to check on FR24 which plane was which and what were their courses, altitudes, etc.

I found a series of photos by A380spotter depicting an interesting case of contrail evolution. Two Cargolux B747s, LX-VCA and LX-SCV, crossed their paths over Heathrow on April 6, 2015 at about 16:00 BST (15:00 UTC):

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I have found these flights on Planefinder.net. I have never used this server before, and I will have to explore it more, as it allows playback of earlier flights than provided by free version of FR24.
The solitary hybrid contrail came from CV436 (CLX436) LX-VCA. Interestingly, this plane was higher at 39,000 ft, but appeared to produce a wider gap between its initially two hybrid contrails, than LX-SCV at 37,000 ft.

Very well done! I especially like the first stereo pair. To my eye that shows the trail confined to a plane as a first approximation. If we define the principle displacements to an x,y plane then displacements in the orthogonal z direction amount to 10 % or less if the x,y displacements. I think it may be misleading to describe the path as a helix. Displacements in the z direction seem to me to be a small and possibly somewhat irregular secondary effect.

Thank you for the explanation, which makes complete sense to me. (I had wondered if the thin trails were perhaps the result of wingtip trails only of which I have one example, but your explanation discounts that. I also agree that KH instability in the atmosphere cannot be the mechanism. I have one example of a trail that I think is disturbed by a kH instability and it looks nothing like the tails above.) Those images are not relevant here, but if you would like to examine them for interest I can put them somewhere.

I have one example of a trail that I think is disturbed by a kH instability and it looks nothing like the tails above.) Those images are not relevant here, but if you would like to examine them for interest I can put them somewhere.

Saw this crazy stuff from Königswinter-Heisterbacherrott, NRW, Germany, through a gap in the cloud cover facing west around 16:00 CET today, i.e. around sunset time – contrast more or less enhanced, no other image manipulations besides cropping. Sadly I didn’t see the vehicle causing this nor what this contrail had looked like initially; airplanes produced several more in this sky path during the brief viewing window – which were perfectly straight.

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This seems to be a busy path in both directions. By the time it could be one of the three A380 that went along it in easterly direction shortly before 15:00 UTC (16:00 CET). The latest one was UAE16:

The camera actually was facing South West, there the Sun was at the time. Its position can be derived from the convergence point of the shadow lines.

This seems to be a busy path in both directions. By the time it could be one of the three A380 that went along it in easterly direction shortly before 15:00 UTC (16:00 CET).

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Could the fact that it is a busy path be significant? A380s are big aircraft and must produce a lot of vortex turbulence. Perhaps the timing just happened to be right for the contrail from one aircraft to get caught up in the turbulence from a preceding one?

The identification of the flight in the time lapse video seems possible, but finding the exact viewpoint in Google Earth to determine the heading of the plane and timing from the sun position may be tricky. Alternatively, a few potential candidate flights could be used as guides in searching for the camera location and viewing directions. My favourite, based on the relative headings and timings of three other contrails in the video is QTR40 A380:

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I have found an approximate viewpoint of the above video. Like another recent video by the same author, it probably was "timelapsed with GoPro Hero3 Black", FOV of which is wider than the ground level view in Google Earth. The QTR40 track downloaded from FlightAware and the Sun position at 15:14 UTC give a nearly perfect match to ground landmarks and other features:

Thus this trail indeed was left by a A380 plane flying a level flight at 39,000 ft. The earth.nullschool.net forecast at 250 hPa suggests there was a 30 km/h tailwind at a slightly lower altitude, but the wind direction was changing from westerly to northerly in this general area.

Update:
Using FR24 and FlightAware, I have identified flights for other contrails in the time lapse video, OK725 at 00:00 (corresponds to 14:48 UTC) and AZ404 at 00:17 (15:22 UTC). Thus the time acceleration factor is about 2 minutes of real time per second of the video and the lifetime of the solitary hybrid contrail was about six minutes.

Attached is a KMZ file with the QTR40 track, the camera viewpoint and the location of a local landmark La Vedetta at the top of the mountain on the left.

Initially, it left a typical hybrid contrail, in one segment of which the contrail strands were separated notably wider than usual:
One of the strands in this segment dissipated rather quickly, whilst the other strand remained for longer and developed a wavy shape before dissipating itself. However, the wave amplitude (in the direction perpendicular to the line of view) was rather weak:

The elapsed time from the formation of this contrail segment to the dissipation of its last strand was about 5 minutes. The last solitary strand outlived a later (younger) segment of the same hybrid contrail, which dissipated in a 'normal' way (through the Crow instability):

The photographer confirmed the location and time of the photo and added that the camera was facing west, or north west. This gives the most likely culprit being the Korean Air Airbus A-380-800 KAL11 from Seoul to Los Angeles that passed over San Fransisco from northwest at about 20:55 UTC (13:55 PDT):

The plane heading almost perfectly was aligned with the wind direction at 250 hPa, but the wind speed was gradually decreasing along the flight track.
Unfortunately, despite it occurred over a big city (San Francisco), I was not able to find more records of the observation of this sinusoidal thin trail, with the exception of the Patrick Roddie's time lapse video:
As his camera's exact location is known, the sun position in the sky can be used to determine the moment (1:15) corresponding the time of the solitary trail observation (21:00 UTC = 14:00 PDT). The Roddie's camera pointing to the South, i.e., in a perpendicular direction, captured a different segment of the KAL11 trail, that might or might not have evolved a sinusoidal shape. At the highest resolution (1440p60 HD) on a big screen, the western end of the trail does look squiggly, but a wide camera angle does not allow to see fine details.

I noticed in the comments section that someone had linked it up to a Qantas A380.

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According to the comment, the author of this collage has also observed and photographed the trail. I agree with this identification, as the positions of the sun and two nascent trail are consistent with the time on FR24 screenshot, but I'd prefer the time of observation being confirmed independently. Regrettably, the photos in the link are stripped off metadata.

A new sighting of this rare contrail from Oxfordshire, UK on November 2, 2016 at about 08:42 UTC. Below is a cropped photo, full version of which can be seen on flickr: https://flic.kr/p/NMeANp
The time and location have been verified by the photo's author.

Once again, the most likely culprit was Airbus A380 (UEA206 A6-EDB) that passed over at FL370 about six minutes earlier.
The ordinary contrail the most likely belongs to EasyJet A319 that passed over at the time of the photo along the same track at FL320.